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Let's say you optimize an experimental geometry of a structure with very large basis sets, strict optimization settings with different XC functionals and after calculating phonons of this structure and you get imaginary frequencies. Does this mean that this structure not stable at all? If so, how they are studied experimentally at all?

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    $\begingroup$ You added the DFT tag, so the following may be valid: the numerical integration of DFT can introduce such artifacts. $\endgroup$ – TAR86 Aug 5 at 5:45
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    $\begingroup$ In terms of a Fourier transform, imaginary frequencies correspond to exponential decay or growth (or sinh/cosh from their linear combinations), I don't know if something similar applies to DFT though $\endgroup$ – Tobias Kienzler Aug 5 at 13:38
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Assuming that all calculation parameters associated with the electronic structure are properly converged, then obtaining imaginary frequencies can mean one of two things.

Physical imaginary frequencies

This situation corresponds to obtaining imaginary frequencies on $\mathbf{q}$-points that are included in the $n_1\times n_2\times n_3$ grid that you explicitly calculate. In this case, the imaginary frequencies are physical and indicate that the structure is not dynamically stable and is not at a minimum of the potential energy landscape, but instead at a saddle point. Distorting the structure along the wave vector associated with the imaginary frequency will allow you to find a lower-energy structure (a recent answer provides details about how to do this).

So how is it possible that you have imaginary modes but the structure is observed experimentally? There could be multiple reasons:

  1. It could be that the structure is at a saddle point of the potential energy surface but at a minimum of the free energy surface at some temperature $T$. In this case, the experimental structure is possibly measured at high temperature and if temperature were lowered experimentally a structural phase transition associated with the imaginary mode would occur. From a computational point of view, you could explore the free energy surface to determine if the imaginary modes disappear at finite temperature by performing anharmonic phonon calculations. A well-known example of this phenomenon are perovskites like BaTiO$_3$, which are cubic at high temperatures experimentally, but if you calculate the phonons you will find imaginary frequencies. This is because at low temperatures this structure transitions from cubic to tetragonal (and at even lower temperatures to other structures).
  2. The above scenario could play out even at $T=0$ K. In this case, rather than anharmonic thermal fluctuations stabilizing the structure, it is quantum fluctuations that do it.
  3. You mention in your question that you have checked different XC functionals. How thorough were you at this? There are examples in which all of LDA, PBE, and other semilocal functionals give imaginary frequencies, but hybrid functionals for example don't. In this case, it would be an issue with the electronic structure method used.

Underconverged imaginary frequencies

This situation corresponds to obtaining imaginary frequencies on $\mathbf{q}$-points that are not included in the $n_1\times n_2\times n_3$ grid that you explicitly calculate. In this case, the imaginary frequencies are probably a result of underconvergence in the phonon part of the calculation and arise from Fourier interpolation performed over the $\mathbf{q}$-point grid that you explicitly calculate to construct frequencies at other $\mathbf{q}$-points.

So how can you address this scenario? The only option is to converge the phonon calculations by performing calculations on finer $\mathbf{q}$-point grids (larger supercells if you are using the finite displacement method).

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You mention phonons, so I assume you are doing periodic structures which I am not entirely familiar with since I study mostly single molecules. Although, when I get imaginary frequencies from optimized geometries of single molecules this typically implies that the isomer in question is not a stable minimum on the potential energy surface (PES). This does not rule out the geometry from being a stable minimum on other multiplicity PES, for example you may have certain isomers give imaginary frequencies on the singlet level but not at the triplet level.

For the case of periodic structures, I can only speculate that if the geometry matches the experimental geometry perfectly then perhaps another parameter does not match the experimental sample.

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